Approximate maximum likelihood estimation for stochastic chemical kinetics

During the last decade stochastic models of networks of chemical reactions have become very popular. The reason is that the assumption that chemical concentrations change deterministically and continuously in time is not always appropriate for cellular processes. In particular, if certain substances in the cell are present in small concentrations the resulting stochastic effects cannot be adequately described by deterministic models. In that case, discrete-state stochastic models are advantageous because they take into account the discrete random nature of chemical reactions. The theory of stochastic chemical kinetics provides a rigorously justified framework for the description of chemical reactions where the effects of molecular noise are taken into account [1]. It is based on discrete-state Markov processes that explicitly represent the reactions as state-transitions between population vectors. When the molecule numbers are large, the solution of the deterministic description of a reaction network and the mean of the corresponding stochastic model agree up to a small approximation error. If, however, species with small populations are involved, then only a stochastic description can provide probabilities of events of interest such as probabilities of switching between different expression states in gene regulatory networks or the distribution of gene expression products. Moreover, even the mean behavior of the stochastic model can largely deviate from the behavior of the deterministic model [2]. In such cases the parameters of the stochastic model rather then the parameters of the deterministic model have to be estimated [3-5].Here, we consider noisy time series measurements of the system state as they are available from wet-lab experiments. Recent experimental imaging techniques such as high-resolution fluorescence microscopy can measure small molecule counts with measurement errors of less than one molecule [6]. We assume that the structure of the underlying re